Under assumption REF  \(U^{\prime }_F\in RV_{\gamma -1}\)  by corollary 1.1.10 in [1]} (up to sign). Then by Potter's inequalities (proposition B.1.9 (5) in [1]}) for any \(\varepsilon , \varepsilon ^{\prime }>0\)  starting from some \(N_0\)  for \(t\ge 1\)  it holds that
\(& (1-\varepsilon ) t^{\gamma -1-\varepsilon ^{\prime }} < \dfrac{U_F^{\prime }\left(\frac{N}{k} t\right)}{U^{\prime }_F\left(\frac{N}{k} \right)}< (1+\varepsilon )t^{\gamma -1+\varepsilon ^{\prime }},\\& (1-\varepsilon ) t^{\gamma -1-\varepsilon ^{\prime }} < \dfrac{U_F^{\prime }\left(\frac{N}{k+s} t\right)}{U^{\prime }_F\left(\frac{N}{k+s} \right)}< (1+\varepsilon )t^{\gamma -1+\varepsilon ^{\prime }}.\) 
